Mechanical Layout of the Indexing Table 


The cam motionlaw is ModSine, 180º index period, and runs at 150RPM. The cycle time is 0.4s with 0.2s motion and 0.2s dwell. We can assume the camshaft runs at constant angular velocity. There is an estimated overall Friction Torque of 2.0Nm in the output transmission referred to the indexer turret. The maximum backlash permitted in the cam track, when manufactured is 0.04mm. The backlash in the gears can wear to 0.1mm before adjustment or replacement. Mass and Inertia of Components
The mass and inertia of each component are calculated as shown in the table above. The final inertia (Combined Total) total in the above table takes into account the gear ratio. The inertia of the slowspeed assembly is reduced by the square of the gear ratio when referred to the indexer turret shaft. ...an illustration of the benefits of a speed reduction! Assessment of Dynamic System.To assess the dynamic response, we estimate the natural vibration frequency of the system, and from that, the PeriodRatio. The gears have a significant mass and inertia in an intermediate position in the transmission, which means that the system will not vibrate with a single, simple frequency, as required by the foregoing dynamic response theory! However, the high inertia of the table and work stations imposes a dominant frequency which gives a good approximation to the theoretical model: this is nearly always the case in practice. But, be aware. RigidityEquations to use : ; Turret Shaft Torsional Rigidity
Second moment of area of shaft is: Thus: Main Shaft Bending Stiffness is: Its equivalent Torsional Rigidity is: Indexer High Speed Shaft Torsional Rigidity Turret Bending Stiffness is: Its equivalent Torsional Rigidity is: The combined rigidity, referred to the indexer turret: [Note that the bending stiffness is so high it could have been ignored]. Approximate Natural FrequencyNatural Period = 1/f = 0.0321 seconds PeriodRatioThe motion period is 0.2 seconds. The PeriodRatio is: n = Motion Period / Natural Period = T / [1/f] = f × T n = 31.17 × 0.2 = 6.234
Nominal Peak Inertia TorqueThe total inertia referred to the indexer is Peak Angular Acceleration: The output stroke of the indexer is: Index Period of the Indexer is: The Coefficient of Acceleration of the ModSine is: Nominal Peak Acceleration is: Peak Inertia Torque after TorqueFactorTorqueFactor We can use this equation to find the TorsionFactor ; The Parameters for the ModSine MotionLaw are: ; ; The PeriodRatio, n. is 6.34 Using the parameters, the TorsionFactor is We must increase the Nominal Peak Inertia Torque by the TorsionFactor: Add Friction TorqueWe must add the Friction Torque for the mechanism, ignoring the backlash impact effect. Include BacklashWe must consider the impact shock load after the transition of backlash... Total Backlash, expressed as an angle (radians), at the Indexer Turret. Gear Teeth, 0.1mm, @ 64mm radius: Indexer Turret, 0.04mm @ 97.4mm radius: Total Backlash The normalized backlash [backlash against angular stroke] Deceleration The natural deceleration of the system due to the deceleration torque on the payload during 'FreeFlight' is: Normalized Deceleration is: This is quite low and can be taken as Zero! Normalized Impact VelocityWith a Normalized Backlash of 0.00251 and Normalized Deceleration of 0.0, then: Normalized Impact Velocity: Real Impact Velocity: Peak Shock TorqueThe Peak Shock Torque on the Turret is: At 40% of the Peak Inertia Torque without Backlash, this is a significant load, and should not be ignored in the design of the mechanism. A safe way of taking it into account is simply to add it to the peak vibration torque (this assumes the two peak torques occur at exactly the same point in the motion, which is quite possible): Peak Torque at Output Shaft.Add the Peak Shock Torque to the Vibration Torque Although it could be argued that this is too pessimistic, it does illustrate that to design the mechanism on the basis of the Nominal Dynamic Torque of 78.41Nm would be underestimated! 